Let be a connected finite undirected graph without loops or multiple edges. For vertices u
and x in, let (u, x) denote the distance between u and x in, ie the length of a shortest path
connecting u and x. Let dd () denote the diameter of, ie the maximal distance between any
two vertices in. We denote by i (u) the set of vertices of at distance i from a vertex u. A graph
is called a distance-regular graph if the numbers ci i 1 (u) 1 (x), ai i (u) 1 (x) and bi i 1 (u) 1 (x)
depend on the distance (u, x) i rather than on individual vertices. When this is the case, we …

This thesis is an exposition of the paper entitled On Distance Regular Graphs with b2= 1 and
Antipodal Covers by Makoto Araya, Akira Hiraki, and Alexander Jurisic. Let T be a Distance
Regular Graph of valency k2. It is shown that if b2= 1, the T is antipodal and one of the
following holds:(1) T is the dodecahedron (2) d= 4 and T is antipodal double cover for a
Strongly Regular Graph with parameters (k, a1, c2)=(n2+ 1, 0, 2) for an integer n not divisible
by four.(3) d= 3 and T is an antipodal cover of a complete graph.